Problem: In the right triangle shown, $AC = BC = 3$. What is $AB$ ? $A$ $C$ $B$ $3$ $3$ $x$
Solution: We know the length of each leg, and want to find the length of the hypotenuse. What mathematical relationship is there between a right triangle's leg and its hypotenuse? We can use either sine (opposite leg over hypotenuse) or cosine (adjacent leg over hypotenuse). Since the two legs of this triangle are congruent, this is a 45-45-90 triangle, and we know what the values of sine and cosine are at each angle of the triangle. Let's try using sine: $A$ $C$ $B$ $3$ $3$ $x$ ${45}^{\circ}$ Sine is opposite over hypotenuse (SOH CAH TOA), so $\sin {45}^{\circ}$ must be $\dfrac{3}{x}$ . We also know that $\sin{45}^{\circ} = \dfrac{\sqrt{2}}{2}$ Solving for $x$ , we get $ x \cdot \sin {45}^{\circ} = 3$ $ x \cdot \dfrac{\sqrt{2}}{2} = 3$ $ x = 3 \cdot \dfrac{2}{\sqrt{2}}$ So, we see that the hypotenuse is $\sqrt{2}$ times as long as each of the legs, since $x = 3 \cdot \sqrt{2}$.